Dishwasher Spray Arm

What is modelled

Water leaves each nozzle as a jet. The momentum it carries away pushes back on the arm, and because the nozzles are tilted the push has a sideways part that turns the arm. The arm accelerates until water drag and bearing friction balance the jet torque.

Equations

• Jet speed v = Cd √(2 Δp / ρ), flow per nozzle Q = A v, A = π d² / 4.
• Thrust per nozzle F = ρ Q v = ρ A v²; its tangential part is F sinα for a tilt α from vertical.
• Jet torque τ = 2 F sinα ∑ ri, nozzles evenly spaced along each arm.
• Spin-up I dω/dt = τ − τf − c ω², with solution ω = ωss tanh(t √(net c) / I) and ωss = √((τ − τf) / c).

Checks run in the page

• Kinetic power of the jets against C_d² x hydraulic power.
• Thrust from momentum flux against ρ A v², and total flow from nozzle count.
• A numerical spin-up integration against the closed form.
• Energy: shaft power stays below hydraulic power, with the efficiency printed.

Honest simplifications

The drag law, the discharge coefficient (0.8), the effective arm mass (2.5 kg lumped with the water inside, to give a visible spin-up) and the even nozzle spacing are illustrative. Real wash arms also have fan-spray nozzles, water paths with pressure loss along the arm, and splash and mist; none of that is modelled, and no cleaning performance is predicted.

• Choose Low pressure: the jet torque falls roughly with pressure and the arm slows.
• Choose Stiff bearing and watch the arm stall when torque drops below friction.
• Drag the tilt slider: a larger tilt gives more sideways push but it saturates near 60 degrees in the graph.